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 Trudy Inst. Mat. i Mekh. UrO RAN, 2011, Volume 17, Number 2, Pages 105–124 (Mi timm701)

Numerical encoding of sampled controls and an approximation metric criterion for the solvability of an aiming game problem

A. V. Kryazhimskiiab

a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow
b International Institute for Applied Systems Analysis, Laxenburg, Austria

Abstract: A game problem of guaranteed aiming in the class of positional strategies is considered for a conflict controlled system with affine scalar controls in the equation of the system. Simplified sampled analogs of quasi-strategies, i.e., of nonanticipating program reactions of the first player to the controls of the second player, are introduced. The nonanticipating property is characterized in metric terms with the use of numerical images (codes) of argument controls and reaction controls. A class of nonanticipating transformations is introduced that is approximately equivalent by the criterion of the solvability of the game problem to the class of positional strategies. The elements of this class as transformations of the numerical codes of controls are characterized by the 1-Lipschitz property. A numerical algorithm for checking the solvability of the problem in this class is described. The complexity order of the algorithm is close to that of the approximation variant of the classical program construction.

Keywords: positional differential games, quasi-strategies, numerical methods.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2012, 276, suppl. 1, S106–S125

Bibliographic databases:

UDC: 517.977.8

Citation: A. V. Kryazhimskii, “Numerical encoding of sampled controls and an approximation metric criterion for the solvability of an aiming game problem”, Trudy Inst. Mat. i Mekh. UrO RAN, 17, no. 2, 2011, 105–124; Proc. Steklov Inst. Math. (Suppl.), 276, suppl. 1 (2012), S106–S125

Citation in format AMSBIB
\Bibitem{Kry11} \by A.~V.~Kryazhimskii \paper Numerical encoding of sampled controls and an approximation metric criterion for the solvability of an aiming game problem \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2011 \vol 17 \issue 2 \pages 105--124 \mathnet{http://mi.mathnet.ru/timm701} \zmath{https://zbmath.org/?q=an:1295.49026} \elib{http://elibrary.ru/item.asp?id=17870027} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2012 \vol 276 \issue , suppl. 1 \pages S106--S125 \crossref{https://doi.org/10.1134/S0081543812020095} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000305482900009} \elib{http://elibrary.ru/item.asp?id=17981874} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84859336754} 

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This publication is cited in the following articles:
1. V. I. Maksimov, “On designing a reconstruction-control algorithm for an ecological-economic model”, Proc. Steklov Inst. Math. (Suppl.), 287, suppl. 1 (2014), 102–115
2. V. N. Ushakov, V. I. Ukhobotov, A. V. Ushakov, G. V. Parshikov, “On solving approach problems for control systems”, Proc. Steklov Inst. Math., 291 (2015), 263–278
3. V. S. Kublanov, V. I. Maksimov, “On feedback-principle control for systems with aftereffect under incomplete phase-coordinate data”, Journal of Mathematical Sciences, 233:4 (2018), 495–513
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